Universality of mesoscopic fluctuations for orthogonal polynomial ensembles

TL;DR

This paper demonstrates the universality of mesoscopic fluctuations in orthogonal polynomial ensembles, showing that measures with similar asymptotic recurrence coefficients exhibit identical fluctuations across scales, with implications for Jacobi ensembles.

Contribution

It establishes a universality result for mesoscopic fluctuations in orthogonal polynomial ensembles based on asymptotic recurrence coefficients and analyzes the Green's function of Jacobi matrices.

Findings

Mesoscopic fluctuations are universal for measures with asymptotic recurrence coefficients.

The convergence rate of recurrence coefficients influences the scale of identical fluctuations.

A Central Limit Theorem is proved for modified Jacobi Unitary Ensembles on all mesoscopic scales.

Abstract

We prove that the fluctuations of mesocopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a Central Limit Theorem for the modified Jacobi Unitary Ensembles on all mesosopic scales.